Page 353 - A Course in Linear Algebra with Applications

9.3: Bilinear Forms 337

Conversely, if q is a quadratic form in x i , . . . , x n, we can
define a corresponding symmetric bilinear form / on R n by
means of the rule

f(X,Y) = ±{q(X + Y)-q(X)-q(Y)}

where X and Y are the column vectors consisting of x\,..., x n
and yi,... ,y n. To see that / is bilinear, first write q(X) =
T
X AX with A symmetric; then we have
T
T
T
f(X, Y) =±{{X + Y) A(X + Y)~ X AX - Y AY}
T T
= \{X AY + Y AX)
T
= X AY,
T
T
T
since X AY = (X AY) T = Y AX. This shows that / is
bilinear.
It is readily seen that the correspondence q —> / just
described is a bijection from quadratic forms to symmetric
n
bilinear forms on R .
Theorem 9.3.3
There is a bijection from the set of quadratic forms in n vari-
n
ables to the set of symmetric bilinear forms on R .
From past experience we would expect to get significant
information about symmetric bilinear forms by using the Spec-
tral Theorem. In fact what is obtained is a canonical or stan-
dard form for such bilinear forms.